Fluid Mechanics for MAP/Fluid Statics
Definitions
editFluid statics is the study of fluids which are either at rest or in rigid body motion with respect to a fixed frame of reference. Rigid body motion means that there is no relative velocity between the fluid particles.
In a fluid at rest, there is no shear stress, i. e. fluid does not deform, but fluid sustains normal stresses.
We can apply Newton's second law of motion to evaluate the reaction of the particle to the applied forces.
Force balance in direction:
We can also say,
Force created by pressure is :
is the vector having the surface area as magnitude and surface normal as direction.
Thus,
Force caused by the pressures opposite to the surface normal.
For a differential fluid element:
Remember Taylor Series expansion:
P is the pressure in the center of the fluid element, therefore the pressure on the surface in direction of is .
Thus,
Thus,
or,
or,
or,
for
Pressure changes only in direction.
Pressure variation in an incompressible and static fluid
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is constant since and are constants.
If we take at the surface, then:
h is measured from the surface.
Transmission of Pressure
editCommunicating containers
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Lets consider two closed containers(which means the free surface pressure could be different than atmospheric pressure) both contain same fluid are connected via a connector valve. When the valve is open, the heights of the fluid columns can give an indication about the pressure in both chamber. For closed container (of course,when we calculate the small 'h', it should be measured at the height of connecting valve for both column distinctively.) for So from the picture above, we can understand that the pressure in the right column is higher than the left column. For open Containers,
If both fluid columns are at the same level so, the depth of the fluid from free surface in both column will be the same. This nice principle was used for Water-based Barometer [1] a.k.a 'Storm Barometer' or 'Goethe Barometer'. Try to see if you understand the device.
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Pressure Measurement Equipments
editBarometer
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From the equations which we derived before , it is also possible to measure the pressure exerted by almost 100 km[2] thick earth atmosphere which is above us. Since the constituents and the density varies over the height of the atmosphere , we will consider a fluid column which have free surface with no atmospheric pressure but connected with a fluid which experience atmospheric pressure like communicating container. Let consider first (from previous section), for atm = kPa water height will be:
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U-tube manometer
editInclined-Tube Manometer
editOther related topics
editBuoyancy and calculation of forces on the submerged surfaces are topics related to fluid statics.